> > Fine for you. My claim says that by digits it is impossible to prove that d > > differs from all rational.

The set of (decimal) digits is }0,1,2,3,4,5,6,7,8,9} And with only those, and no knowledge of digit positions, only one digit numbers are distinguishable from all other numbers.

> > That means up to every finite digit

But since no single digit (or even finite set of digits) determines any rational numbers entire value, a single digit value can only prove it unequal to infinitely many other numbers, but not prove it equal to any of them.

> > And if you acknowledge that d is nothing more than its digits, then there > > are infinitely many rational identical with d.

But nobody sensible would ever "acknowledge" such a blatant falsehood.

Actually it is far more that just digits, it is the set of pairings of digit values with digit positions which is required to determine value.

If digit positions were not relevant then 0.12 and 0.21 would be equal, as they have the set of same digit values.